The process of letting delta t get smaller and smaller
so that one comes up with an instantaneous rate of change
can also be viewed graphically.
This may give you another way to picture this process
and, more generally, to think about what
differential calculus is all about.
So, as a starting point suppose we have a curve -
some function that may be looks like this,
and this curve could represent the growth of some process -
who knows, it doesn't matter - it's just a curve,
and then somebody asked you to find the slope of the curve -
that would be like the instantaneous velocity,
or instantaneous speed, or growth rate.
You might say, well I don't know how to find the slope of a curve -
a slope is a property of a straight line
and the curve isn't a straight line
but aha, then you might get an idea.
You look at the curve - I'm looking at it now,
well it looks like a curve - I can see it bending
however, if I look really, really close -
I zoom in until my eye is almost against it,
it looks like a straight line
and I know how to find the slope of a straight line,
so, to find the slope at a point,
I just zoom in until it looks sufficiently line-like
and then I find the slope - no big deal.
So this idea - that, if you take a curve
and you zoom in on it, that it looks like a line.
and you can calculate the slope of the line,
that's the one big idea
behind all of differential calculus.
So, in the first term of a calculus sequence
one typically spends most of the time
defining this a lot more carefully,
and then exploring the consequences of it.
Here's another way to illustrate the fact
that zooming in on a curve
makes it look like a straight line.
So, here's a curved function,
viewed from this distance,
it definitely does not look straight,
and now imagine zooming in -
so I'm moving the function
closer and closer to the camera
and you should see that the curve starts to
look more and more like a straight line.
If it doesn't look sufficiently straight you could zoom in more.
I'm limited because eventually I'm going to hit
the lens cover of the camera and I can't zoom in any more
but if you zoom in more and more, and more and more,
the line looks straighter and straighter and straighter,
and you can try this at home -
draw a curve on a piece of paper, zoom in
and you'll see that it really does look like a straight line,
so differential calculus works.